Exam 19: Multiple Regression

In multiple regression with k independent variables, the t-tests of the individual coefficients allow us to determine whether $\beta _ { i } \neq 0$ (for i = 1, 2, …, k), which tells us whether a linear relationship exists between $x _ { i }$ and y.

A statistics professor investigated some of the factors that affect an individual student's final grade in his or her course. He proposed the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . Where: y = final mark (out of 100). $x _ { 1 }$ = number of lectures skipped. $x _ { 2 }$ = number of late assignments. $X _ { 3 }$ = mid-term test mark (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below. THE REGRESSION EQUATION IS $= 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predictor Coef StDev Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $\mathrm{S}=13.74 \quad \mathrm{R}-\mathrm{Sq}=30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 Do these data provide enough evidence to conclude at the 5% significance level that the final mark and the number of skipped lectures are linearly related?

For a set of 30 data points, Excel has found the estimated multiple regression equation to be = -8.61 + 22x₁ + 7x₂ + 28x₃, and has listed the t statistic for testing the significance of each regression coefficient. Using the 5% significance level for testing whether $\beta$ ₃ = 0, the critical region will be that the absolute value of the t statistic for $\beta$ ₃ is greater than or equal to:

In a multiple regression analysis involving 6 independent variables and a sample of 19 data points the total variation in y is SS_y = 900 and the amount of variation in y that is explained by the variations in the independent variables is SSR = 600. The value of the F-test statistic for this model is:

To test the validity of a multiple regression model involving 2 independent variables, the null hypothesis is that:

An actuary wanted to develop a model to predict how long individuals will live. After consulting a number of physicians, she collected the age at death (y), the average number of hours of exercise per week ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $X _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev T Constant 55.8 11.8 4.729 1.79 0.44 4.068 -0.021 0.011 -1.909 -0.016 0.014 -1.143 S = 9.47 R-Sq = 22.5%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 936 312 3.477 Error 36 3230 89.722 Total 39 4166 Is there enough evidence at the 5% significance level to infer that the number of points by which the individual's blood pressure exceeded the recommended value and the age at death are negatively linearly related?

A multiple regression model involves8 independent variables and 32 observations. If we want to test at the 5% significance level the parameter $\beta _ { 4 }$ , the critical value will be:

The adjusted multiple coefficient of determination is adjusted for the number of independent variables and the sample size.

For a multiple regression model with n = 35 and k = 4, the following statistics are given: SS_y = 500 and SSE = 100. The coefficient of determination is:

In a multiple regression model, the probability distribution of the error variable $\varepsilon$ is assumed to be:

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the number of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are 4 and 20, respectively.

The adjusted multiple coefficient of determination is adjusted for the:

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

A statistician estimated the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \xi$ , with 45 observations. The computer output is shown below. However, because of a printer malfunction, some of the results are not shown. These are indicated by the boldface letters a to l. Fill in the missing results (up to three decimal places). Predictor Coef StDev T Constant 2.794 a 6.404 b 0.007 -0.025 0.383 0.072 c S = d R-Sq = e. ANALYSIS OF VARIANCE Source of Variation Regression f i j l Error g 11.884 k Total h 26.887

In a multiple regression analysis involving 50 observations and 5 independent variables, SST = 475 and SSE = 71.25. The multiple coefficient of determination is 0.85.

In multiple regression, the Durbin-Watson test is used to determine if there is autocorrelation in the regression model

Excel and Minitab print a second $R ^ { 2 }$ statistic, called the coefficient of determination adjusted for degrees of freedom, which has been adjusted to take into account the sample size and the number of independent variables.

In testing the significance of a multiple regression model in which there are three independent variables, the null hypothesis is Ho: β₀ = β₁ = β₂ = β₃.

Which of the following statements is not true?

An actuary wanted to develop a model to predict how long individuals will live. After consulting a number of physicians, she collected the age at death (y), the average number of hours of exercise per week ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $X _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev T Constant 55.8 11.8 4.729 1.79 0.44 4.068 -0.021 0.011 -1.909 -0.016 0.014 -1.143 S = 9.47 R-Sq = 22.5%. ANALYSIS OF VARIANCE Source olf Variation df SS MS F Regression 3 936 312 3.477 Error 36 3230 89.722 Total 39 4166 Interpret the coefficient $b _ { 2 }$ .